1. Introduction

Intersubband optical nonlinearities in semiconductor heterostructures, particularly InGaAs-AlInAs on InP, have gained much interest since the observation of intersubband second harmonic generation (SHG) in InGaAs/AlInAs coupled quantum wells by Sirtori et al. in 1991 [1]. This observation was closely followed by the realization of intersubband third harmonic generation (THG) in this material system [2]. Thereafter, the focus has been to optimize the second and third order nonlinear susceptibilities, χ ⁽²⁾ and χ ⁽³⁾, respectively. Since these susceptibilities are proportional to the product of the dipole matrix elements of the electronic intersubband transitions involved [3], [4], [5], the most popular technique used to optimize the nonlinear susceptibilities has been to simply maximize this product {z₁₂* z₂₃* z₃₁ for χ ⁽²⁾ and z₁₂* z₂₃*z₃₄* z₄₁ for χ ⁽³⁾}, where z_ij is the optical dipole matrix element between energy levels i and j, while keeping the energy levels in near resonance with the pump radiation. Even with the maximization of this product there are problems that render such intersubband transitions impractical for nonlinear optical generation. The incomplete list of these problems includes (i) an inefficient coupling of the external pump radiation to the intersubband transitions, (ii) a low overlap of the external pump beam and the generated nonlinear signal with the nonlinear region, and (iii) the resonant absorption of the external pump.

Here we present a Quantum Cascade (QC) laser having an active region designed for fundamental pump radiation emission and simultaneous third order nonlinear emission. As in [6] and [7], an integrated device has been designed in which the quantum wells (QWs) in the active region of a QC laser simultaneously function as a pump laser and a nonlinear oscillator. This design ensures efficient coupling of the pump photon energy ħω to the resonant transitions of the nonlinear element and maximal overlap of the interacting waveguide modes with an active nonlinear region. The problem of pump absorption is also solved since the pump is actually generated in the nonlinear medium. We observed fundamental light emission at 11.1 µm wavelength and SHG and THG emission at ~5.4 µm and 3.7 µm, respectively.

2. Experiment

The sample was grown by molecular beam epitaxy in the InGaAs/AlInAs system lattice matched to InP substrate. Fifty-two periods of QC laser active regions interleaved with injector regions are sandwiched between two layers of InGaAs, a 1.3 µm thick one below and a 1.2 µm thick one above, both doped with Si to n_Si~5×10¹⁶ cm^-3. All these layers constitute the waveguide core. The bottom cladding is provided by the low-doped (n_Si~3×10¹⁷ cm^-3) InP substrate. The top cladding is made of an inner 1.2 µm and an outer 0.8 µm thick AlInAs layer, doped to n_Si~1×10¹⁷ cm^-3 and n_Si~2×10¹⁷ cm^-3, respectively, capped by a highly doped (n_Si~3×10¹⁸ cm^-3) 0.8 µm thick InGaAs layer [8]. Short transition regions are grown between all bulk-like layers for better carrier transport. This waveguide has been designed for low optical loss at the fundamental frequency; however, phase matching considerations have not been included, which explains the low yield of devices with significant nonlinear light emission. True phase-matching in QC-lasers with integrated nonlinear regions has later been demonstrated for SHG [9] using waveguide modes of different transverse order. The same technique can be applied to THG devices [10].

Figure 1 shows the conduction band diagram for one active region sandwiched between two injector regions calculated using Schrödinger’s equation under an applied bias of 42 kV/cm. This field is considerably higher than the turn-on electric field calculated and measured as 30 kV/cm, and more appropriate to describe the structure under high pumping conditions, where we also measure the highest optical power. The structure is designed such that energy levels 1, 2, and 3 facilitate lasing at the fundamental frequency and the nonlinear processes are supported by energy levels 2, 3, 4, and 5. By design, pump laser action is achieved by population inversion between levels 3 and 2 in the three-well active region. A longitudinal optical phonon scattering time of 6.61 ps is calculated for the diagonal transition from energy level 3 to 2; and a significantly faster relaxation (0.64 ps) from level 2 to level 1. The energy spacings ΔE₂₃, ΔE₃₄, and ΔE₄₅ are 98 meV, 104 meV, and 90 meV, respectively. The values under laser operation are, however, strongly dependent on the actual local electric field, due to the diagonal nature of most transitions in the active region, and the ensuing Stark tuning. Since our target nonlinear light generation occurs as a result of the 2-3-4-5 cascade, we are most interested in the corresponding optical dipole matrix elements. They are calculated to be z₂₃=14.6 Å, z₃₄=21.5 Å, z₄₅=33.2 Å, and z₂₅=7.0 Å, yielding a significantly greater dipole matrix element product than that of previous asymmetric coupled QW designs intended for intersubband SHG or THG in the AlInAs/GaInAs material system [2], [6], [7]. (Such comparisons are, however, of limited value as optical dipole moments are generally larger for smaller laser and resonance energies ΔE_ij, as it is the case here.)

 

Fig. 1. Conduction band energy diagram for one active region between two injector regions. The significant energy levels inside the active region are labeled 1 to 5. The thicknesses in nanometers of the InGaAs QW’s and AlInAs barriers of one period of injector and active region are from right to left: 3.7/2.1/3.0/2.1/3.5/2.1/3.4/3.6/3.1/1.2/6.4/1.3/4.7/2.6, the barriers are indicated in bold font and the underlined layers are doped to n_Si=2.5×10¹⁶ cm^-3. The yellow bars indicate the levels involved in THG; the red bars and arrow indicates the laser transition.

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Fig. 2. Emission spectra of a QC laser with integrated third-order nonlinear oscillator in the active region. Part (A) shows the spectrum of the fundamental light emission at a wavelength of 11.1 µm. The spectrum was taken with a MCT detector. Part (B) shows the SHG and THG light emission from the same device. The SHG and THG are at wavelengths of 5.4 µm and 3.7 µm, respectively. The spectra were taken with an InSb detector.

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Figure 2 shows the emission spectra from the integrated device taken at a temperature of 6 K. Part (A) shows a Fabry-Perot spectrum characteristic for the radiation at the pump wavelength. Part (B) shows the THG signal at λ~3.7 µm from the same device that displays a fundamental emission at λ~11.1 µm. The laser was operated in pulsed mode with 50 ns pulses on a low (<1%) duty cycle, and at high peak current to obtain a large optical power. The nonlinear and linear emission spectra were taken with a calibrated, amplified, liquid nitrogen cooled InSb photovoltaic detector and a liquid nitrogen cooled HgCdTe (MCT) detector, respectively. The collection efficiency of the optics is near unity. The spectral response of the InSb detector fitted with a short-pass filter is shown as the dashed line in Fig. 2. The small broad background emission can be attributed to background luminescence from electrons excited to high energy levels.

The laser spectrum as shown in Fig. 2(A) is broad, which is not uncommon for a Fabry-Perot type QC laser operated at peak power. The peak intensity wavelength is around 11.1 µm. As the nonlinear light generation increases superlinearly with the pump power, we focus at this value of the wavelength. The measured THG signal is peaked at 3.7 µm wavelength, exactly as expected of the harmonic. This gives us confidence that what is measured is indeed THG rather than spontaneous emission from electrons excited into level 5 through multiple, sequential absorption of pump photons. The SHG light measured around 5.4 µm wavelength overlaps with the firm cut-off of the InSb detector and short-pass filter. As a result, its actual peak wavelength was not accurately determined. The laser spectrum in part (A) is measured with high resolution, in part (B) with low resolution, using the step-scan mode of the Fourier Transform spectrometer.

3. Theory

In the weak-field approximation, the nonlinear polarization amplitude at the third harmonic is given by [6]

Here E _z(ω) is the z-component of the electric field at fundamental frequency ω; z is the coordinate in the growth direction [001], n _2,3,4,5 are the normalized electron populations of states 2,3,4,5, Γ _mn =γ_mn +i(ω_mn -ω), where ω_mn and γ_mn are the frequency and the total linewidth of the transition between states m and n. The expression for the amplitude σ₅₂ of the off-diagonal element of the density matrix has been derived from the general set of equations, namely Eqs. (7),(8) of reference 6, assuming that the intensities of all fields are lower than the saturation intensities for all optical transitions involved in the THG process. Effects of saturation and power broadening have been neglected in equation 1. This assumption is justified in the present experiment, although in principle, the density matrix formalism allows one to calculate the nonlinear optical response for arbitrarily strong field intensities.

Using the calculated dipole moments and electron density N_e~5×10¹⁵ cm^-3, assuming that all population is in state 3, neglecting all detunings (i.e. Γ _mn =γ_mn ), and assuming all gammas (γ_mn) equal to 10 meV except γ ₅₂~20 meV, we arrive at χ ⁽³⁾=P(3ω)/${{\mathrm{E}}_{\mathrm{z}}}^3$(ω)~7×10^-8 esu. This is because we managed to achieve a very large product of the four dipole moments. The nonlinear polarization at the second harmonic for the same set of electron states 2,3,4,5 interacting with the laser field generated on the transition 3-2 is given by equation 10 of reference 6. We use this equation to calculate the second-order susceptibility χ ⁽²⁾ and find that the ratio of third-order and second-order susceptibilities for the same structure scales as

This relatively high ratio implies that direct generation of a third harmonic by the χ ⁽³⁾ resonant nonlinearity yields higher power than the mixing of the fundamental laser radiation with the signal at a second harmonic.

The nonlinear power at a third harmonic is given as

where α ₃ stands for the total losses of a given cavity mode at wavelength λ₃ of the third harmonic, L the cavity length, R _1,3 and µ _1,3 are reflection factors of a cavity and effective refractive indices of modes at wavelengths λ_1,3, respectively; Δk=k _3ω-3k_ω =3ω(µ ₃-µ ₁)/c and W ₁ is the power in the fundamental mode. The nonlinear overlap factor ∑ is given by

where H _ω,3ω(x, z) are transverse profiles of TM modes of the waveguide at frequencies ω and 3ω. The waveguide axis is along y-coordinate.

Equations (3) and (4) have been obtained similarly to the derivation of the nonlinear signal power at the second harmonic outlined in [6]. Namely, the magnetic field amplitude H_x(r) of a given TM-mode with frequency 3ω and longitudinal wavenumber k=3ωµ/c was represented as a product of function A(y)exp(iky), where the complex function A(y) varies slowly with coordinate y along the waveguide direction, and the transverse distribution H _3ω(x, z), which satisfies the transverse Helmholtz equation for the “cold” waveguide. Substituting the above representation for H_x into Maxwell’s equations, using the orthogonality of the transverse functions H for modes of the different order, and solving the resulting differential equation for the amplitude A _3ω, we finally arrive at Eqs. (3) and (4). As is clear from the first integral in Eq. (4), ∑ is a measure of the overlap of the transverse profiles of the interacting electromagnetic modes with the nonlinear region. Its value is inversely proportional to the effective interaction area.

Substituting all numbers in Eq. 3, using calculated mode profiles and effective indices, we obtain a third order nonlinear power of order 30–50 nW for the fundamental power of 100 mW, depending on the ridge width, which in the experiments ranged from ~10–20 µm. A power level in the low 10’s nW regime is consistent with our measured detector signal.

4. Conclusion

In conclusion, we have observed THG from intersubband transitions within a QC laser active region that functions simultaneously as a fundamental pump and nonlinear light source. Continued work is needed in order to improve the conversion efficiency of linear to nonlinear radiation in this structure, which will increase the THG power. We believe that the incorporation of phase matching into the waveguide design will significantly increase the device performance. This work demonstrates that QC-lasers can be designed with integrated optical nonlinearities with large χ ⁽³⁾ for THG.

The work at Bell Laboratories was in part supported by DARPA/US ARO under contract number DAAD19-00-C-0096. A.B. acknowledges the support from the TAMU TITF Initiative.